Loop Quantum Cosmology III: Wheeler–DeWitt Operators

In the framework of loop quantum cosmology anomaly free quantizations of the Hamiltonian constraint for Bianchi class A, locally rotationally symmetric and isotropic models are given. Basic ideas of the construction in (non-symmetric) loop quantum gravity can be used, but there are also further inputs because the special structure of symmetric models has to be respected by operators. In particular, the basic building blocks of the homogeneous models are point holonomies rather than holonomies necessitating a new regularization procedure. In this respect, our construction is applicable also for other (non-homogeneous) symmetric models, e.g. the spherically symmetric one.